Clustering and Classification
The list of materials and links related to clustering and classification can be found below.
course slides by Emma Kämäräinen
DataCamp exercise
RStudio Exercise 4
After solving the DataCamp exercise and going through the embedded links, I got a general overview on the topic. In the following sections, I will prepare a report based on the exercise instructions.
Data
In this exercise, I will be using Boston data from MASS package.
library(MASS)
data(Boston)
str(Boston)## 'data.frame': 506 obs. of 14 variables:
## $ crim : num 0.00632 0.02731 0.02729 0.03237 0.06905 ...
## $ zn : num 18 0 0 0 0 0 12.5 12.5 12.5 12.5 ...
## $ indus : num 2.31 7.07 7.07 2.18 2.18 2.18 7.87 7.87 7.87 7.87 ...
## $ chas : int 0 0 0 0 0 0 0 0 0 0 ...
## $ nox : num 0.538 0.469 0.469 0.458 0.458 0.458 0.524 0.524 0.524 0.524 ...
## $ rm : num 6.58 6.42 7.18 7 7.15 ...
## $ age : num 65.2 78.9 61.1 45.8 54.2 58.7 66.6 96.1 100 85.9 ...
## $ dis : num 4.09 4.97 4.97 6.06 6.06 ...
## $ rad : int 1 2 2 3 3 3 5 5 5 5 ...
## $ tax : num 296 242 242 222 222 222 311 311 311 311 ...
## $ ptratio: num 15.3 17.8 17.8 18.7 18.7 18.7 15.2 15.2 15.2 15.2 ...
## $ black : num 397 397 393 395 397 ...
## $ lstat : num 4.98 9.14 4.03 2.94 5.33 ...
## $ medv : num 24 21.6 34.7 33.4 36.2 28.7 22.9 27.1 16.5 18.9 ...dim(Boston)## [1] 506 14The Boston data was collected to study the housing values in the suburbs of Boston. The table contains 506 observations for 14 different variables. The descriptions for each of the 14 variables are listed below.
| Variables | Description |
|---|---|
| crim | per capita crime rate by town. |
| zn | proportion of residential land zoned for lots over 25,000 sq.ft. |
| indus | proportion of non-retail business acres per town. |
| chas | Charles River dummy variable (= 1 if tract bounds river; 0 otherwise). |
| nox | nitrogen oxides concentration (parts per 10 million). |
| rm | average number of rooms per dwelling. |
| age | proportion of owner-occupied units built prior to 1940. |
| dis | weighted mean of distances to five Boston employment centres. |
| rad | index of accessibility to radial highways. |
| tax | full-value property-tax rate per $10,000. |
| ptratio | pupil-teacher ratio by town. |
| black | 1000(Bk - 0.63)^2 where Bk is the proportion of blacks by town. |
| lstat | lower status of the population (percent). |
| medv | median value of owner-occupied homes in $1000s. |
Data Summary
Now, let’s look at the summary of the boston data in the form of table (instead of default layout) using pandoc.table function of pander package.
library(pander)
pandoc.table(summary(Boston), caption = "Summary of Boston data", split.table = 120)##
## -----------------------------------------------------------------------------------------------------------------------
## crim zn indus chas nox rm age
## ------------------ ---------------- --------------- ----------------- ---------------- --------------- ----------------
## Min. : 0.00632 Min. : 0.00 Min. : 0.46 Min. :0.00000 Min. :0.3850 Min. :3.561 Min. : 2.90
##
## 1st Qu.: 0.08204 1st Qu.: 0.00 1st Qu.: 5.19 1st Qu.:0.00000 1st Qu.:0.4490 1st Qu.:5.886 1st Qu.: 45.02
##
## Median : 0.25651 Median : 0.00 Median : 9.69 Median :0.00000 Median :0.5380 Median :6.208 Median : 77.50
##
## Mean : 3.61352 Mean : 11.36 Mean :11.14 Mean :0.06917 Mean :0.5547 Mean :6.285 Mean : 68.57
##
## 3rd Qu.: 3.67708 3rd Qu.: 12.50 3rd Qu.:18.10 3rd Qu.:0.00000 3rd Qu.:0.6240 3rd Qu.:6.623 3rd Qu.: 94.08
##
## Max. :88.97620 Max. :100.00 Max. :27.74 Max. :1.00000 Max. :0.8710 Max. :8.780 Max. :100.00
## -----------------------------------------------------------------------------------------------------------------------
##
## Table: Summary of Boston data (continued below)
##
##
## ------------------------------------------------------------------------------------------------------------------
## dis rad tax ptratio black lstat medv
## ---------------- ---------------- --------------- --------------- ---------------- --------------- ---------------
## Min. : 1.130 Min. : 1.000 Min. :187.0 Min. :12.60 Min. : 0.32 Min. : 1.73 Min. : 5.00
##
## 1st Qu.: 2.100 1st Qu.: 4.000 1st Qu.:279.0 1st Qu.:17.40 1st Qu.:375.38 1st Qu.: 6.95 1st Qu.:17.02
##
## Median : 3.207 Median : 5.000 Median :330.0 Median :19.05 Median :391.44 Median :11.36 Median :21.20
##
## Mean : 3.795 Mean : 9.549 Mean :408.2 Mean :18.46 Mean :356.67 Mean :12.65 Mean :22.53
##
## 3rd Qu.: 5.188 3rd Qu.:24.000 3rd Qu.:666.0 3rd Qu.:20.20 3rd Qu.:396.23 3rd Qu.:16.95 3rd Qu.:25.00
##
## Max. :12.127 Max. :24.000 Max. :711.0 Max. :22.00 Max. :396.90 Max. :37.97 Max. :50.00
## ------------------------------------------------------------------------------------------------------------------After getting a summary of the data, it’s worthwhile to get graphical representation. This time we will make a correlogram, a graphical representation of coorelation matrix. The corrplot function of corrplot package wll be used.
library(corrplot)## corrplot 0.84 loadedlibrary(dplyr)##
## Attaching package: 'dplyr'## The following object is masked from 'package:MASS':
##
## select## The following objects are masked from 'package:stats':
##
## filter, lag## The following objects are masked from 'package:base':
##
## intersect, setdiff, setequal, unioncorr_boston<-cor(Boston) %>% round(2)
pandoc.table(corr_boston, split.table = 120)##
## -------------------------------------------------------------------------------------------------------------------------------
## crim zn indus chas nox rm age dis rad tax ptratio black lstat medv
## ------------- ------- ------- ------- ------- ------- ------- ------- ------- ------- ------- --------- ------- ------- -------
## **crim** 1 -0.2 0.41 -0.06 0.42 -0.22 0.35 -0.38 0.63 0.58 0.29 -0.39 0.46 -0.39
##
## **zn** -0.2 1 -0.53 -0.04 -0.52 0.31 -0.57 0.66 -0.31 -0.31 -0.39 0.18 -0.41 0.36
##
## **indus** 0.41 -0.53 1 0.06 0.76 -0.39 0.64 -0.71 0.6 0.72 0.38 -0.36 0.6 -0.48
##
## **chas** -0.06 -0.04 0.06 1 0.09 0.09 0.09 -0.1 -0.01 -0.04 -0.12 0.05 -0.05 0.18
##
## **nox** 0.42 -0.52 0.76 0.09 1 -0.3 0.73 -0.77 0.61 0.67 0.19 -0.38 0.59 -0.43
##
## **rm** -0.22 0.31 -0.39 0.09 -0.3 1 -0.24 0.21 -0.21 -0.29 -0.36 0.13 -0.61 0.7
##
## **age** 0.35 -0.57 0.64 0.09 0.73 -0.24 1 -0.75 0.46 0.51 0.26 -0.27 0.6 -0.38
##
## **dis** -0.38 0.66 -0.71 -0.1 -0.77 0.21 -0.75 1 -0.49 -0.53 -0.23 0.29 -0.5 0.25
##
## **rad** 0.63 -0.31 0.6 -0.01 0.61 -0.21 0.46 -0.49 1 0.91 0.46 -0.44 0.49 -0.38
##
## **tax** 0.58 -0.31 0.72 -0.04 0.67 -0.29 0.51 -0.53 0.91 1 0.46 -0.44 0.54 -0.47
##
## **ptratio** 0.29 -0.39 0.38 -0.12 0.19 -0.36 0.26 -0.23 0.46 0.46 1 -0.18 0.37 -0.51
##
## **black** -0.39 0.18 -0.36 0.05 -0.38 0.13 -0.27 0.29 -0.44 -0.44 -0.18 1 -0.37 0.33
##
## **lstat** 0.46 -0.41 0.6 -0.05 0.59 -0.61 0.6 -0.5 0.49 0.54 0.37 -0.37 1 -0.74
##
## **medv** -0.39 0.36 -0.48 0.18 -0.43 0.7 -0.38 0.25 -0.38 -0.47 -0.51 0.33 -0.74 1
## -------------------------------------------------------------------------------------------------------------------------------corrplot(corr_boston, method = "circle", tl.col = "black", type = "upper" , tl.cex = 0.9 )
In the graph, positive correlations are displayed in blue and negative correlations in red color with intensity of the color and circle size being proportional to the correlation coefficients.
Data Standardization
boston_scaled<-scale(Boston)
pandoc.table(summary(Boston), split.table = 120)##
## -----------------------------------------------------------------------------------------------------------------------
## crim zn indus chas nox rm age
## ------------------ ---------------- --------------- ----------------- ---------------- --------------- ----------------
## Min. : 0.00632 Min. : 0.00 Min. : 0.46 Min. :0.00000 Min. :0.3850 Min. :3.561 Min. : 2.90
##
## 1st Qu.: 0.08204 1st Qu.: 0.00 1st Qu.: 5.19 1st Qu.:0.00000 1st Qu.:0.4490 1st Qu.:5.886 1st Qu.: 45.02
##
## Median : 0.25651 Median : 0.00 Median : 9.69 Median :0.00000 Median :0.5380 Median :6.208 Median : 77.50
##
## Mean : 3.61352 Mean : 11.36 Mean :11.14 Mean :0.06917 Mean :0.5547 Mean :6.285 Mean : 68.57
##
## 3rd Qu.: 3.67708 3rd Qu.: 12.50 3rd Qu.:18.10 3rd Qu.:0.00000 3rd Qu.:0.6240 3rd Qu.:6.623 3rd Qu.: 94.08
##
## Max. :88.97620 Max. :100.00 Max. :27.74 Max. :1.00000 Max. :0.8710 Max. :8.780 Max. :100.00
## -----------------------------------------------------------------------------------------------------------------------
##
## Table: Table continues below
##
##
## ------------------------------------------------------------------------------------------------------------------
## dis rad tax ptratio black lstat medv
## ---------------- ---------------- --------------- --------------- ---------------- --------------- ---------------
## Min. : 1.130 Min. : 1.000 Min. :187.0 Min. :12.60 Min. : 0.32 Min. : 1.73 Min. : 5.00
##
## 1st Qu.: 2.100 1st Qu.: 4.000 1st Qu.:279.0 1st Qu.:17.40 1st Qu.:375.38 1st Qu.: 6.95 1st Qu.:17.02
##
## Median : 3.207 Median : 5.000 Median :330.0 Median :19.05 Median :391.44 Median :11.36 Median :21.20
##
## Mean : 3.795 Mean : 9.549 Mean :408.2 Mean :18.46 Mean :356.67 Mean :12.65 Mean :22.53
##
## 3rd Qu.: 5.188 3rd Qu.:24.000 3rd Qu.:666.0 3rd Qu.:20.20 3rd Qu.:396.23 3rd Qu.:16.95 3rd Qu.:25.00
##
## Max. :12.127 Max. :24.000 Max. :711.0 Max. :22.00 Max. :396.90 Max. :37.97 Max. :50.00
## ------------------------------------------------------------------------------------------------------------------Next, we will create quantile vector for crime
boston_scaled<- data.frame(boston_scaled)
qvc<-quantile(boston_scaled$crim)
crime <- cut(boston_scaled$crim, breaks = qvc, label = c("low", "med_low", "med_high", "high"), include.lowest = TRUE)
boston_scaled <- dplyr::select(boston_scaled, -crim)
boston_scaled<-data.frame(boston_scaled, crime)
table(boston_scaled$crime)##
## low med_low med_high high
## 127 126 126 127Linear Discriminant Analysis
library(MASS)
n<-nrow(boston_scaled)
ind <- sample(n, size = n*0.8)
train <- boston_scaled[ind,]
test <- boston_scaled[-ind,]
lda.fit <- lda(crime ~ ., data = train)
# target classes as numeric
classes <- as.numeric(train$crime)
# plot the lda results
plot(lda.fit, dimen = 2, col = classes, pch = classes)
Class Prediction
crime_cat<-test$crime
test<-dplyr::select(test, -crime)
lda.pred<-predict(lda.fit, newdata = test)
table(correct = crime_cat, predicted = lda.pred$class)## predicted
## correct low med_low med_high high
## low 8 8 2 0
## med_low 6 20 7 0
## med_high 0 8 18 1
## high 0 0 0 24K-means Clustering
data(Boston)
boston_scaled1<-as.data.frame(scale(Boston))
dist_eu<-dist(boston_scaled1)
summary(dist_eu)## Min. 1st Qu. Median Mean 3rd Qu. Max.
## 0.1343 3.4625 4.8241 4.9111 6.1863 14.3970head(boston_scaled1)## crim zn indus chas nox rm
## 1 -0.4193669 0.2845483 -1.2866362 -0.2723291 -0.1440749 0.4132629
## 2 -0.4169267 -0.4872402 -0.5927944 -0.2723291 -0.7395304 0.1940824
## 3 -0.4169290 -0.4872402 -0.5927944 -0.2723291 -0.7395304 1.2814456
## 4 -0.4163384 -0.4872402 -1.3055857 -0.2723291 -0.8344581 1.0152978
## 5 -0.4120741 -0.4872402 -1.3055857 -0.2723291 -0.8344581 1.2273620
## 6 -0.4166314 -0.4872402 -1.3055857 -0.2723291 -0.8344581 0.2068916
## age dis rad tax ptratio black
## 1 -0.1198948 0.140075 -0.9818712 -0.6659492 -1.4575580 0.4406159
## 2 0.3668034 0.556609 -0.8670245 -0.9863534 -0.3027945 0.4406159
## 3 -0.2655490 0.556609 -0.8670245 -0.9863534 -0.3027945 0.3960351
## 4 -0.8090878 1.076671 -0.7521778 -1.1050216 0.1129203 0.4157514
## 5 -0.5106743 1.076671 -0.7521778 -1.1050216 0.1129203 0.4406159
## 6 -0.3508100 1.076671 -0.7521778 -1.1050216 0.1129203 0.4101651
## lstat medv
## 1 -1.0744990 0.1595278
## 2 -0.4919525 -0.1014239
## 3 -1.2075324 1.3229375
## 4 -1.3601708 1.1815886
## 5 -1.0254866 1.4860323
## 6 -1.0422909 0.6705582Inspired by this R-blogger post and this Stackoverflow question
First we start with random cluster number. Let’s start with k=4 and apply k-means on the data.
#Let us apply kmeans for k=4 clusters
kmm = kmeans(boston_scaled1,6,nstart = 50 ,iter.max = 15) #we keep number of iter.max=15 to ensure the algorithm converges and nstart=50 to ensure that atleat 50 random sets are choosen
kmm## K-means clustering with 6 clusters of sizes 119, 81, 185, 48, 39, 34
##
## Cluster means:
## crim zn indus chas nox rm
## 1 0.4622787 -0.4872402 1.1821100 -0.2723291 1.0714202 -0.5414749
## 2 -0.4124983 1.9281718 -1.0907891 -0.2237229 -1.1507665 0.5969852
## 3 -0.3797266 -0.3458615 -0.3166917 -0.2723291 -0.4087995 -0.2997470
## 4 -0.3766805 -0.1326105 -0.8968664 -0.2723291 -0.2431916 1.5614700
## 5 1.8841530 -0.4872402 1.0205262 -0.2723291 1.0253723 -0.3279585
## 6 -0.1985497 -0.2602436 0.2799956 3.6647712 0.3830784 0.2756681
## age dis rad tax ptratio black
## 1 0.81621486 -0.8344560 1.029393697 1.1774470 0.61868734 0.1328957
## 2 -1.40494992 1.5631737 -0.624570346 -0.5824419 -0.68600489 0.3520290
## 3 -0.15958722 0.1869013 -0.592013203 -0.5838115 0.08645442 0.2740981
## 4 0.06875999 -0.2891339 -0.520091751 -0.8271402 -1.03318239 0.3543957
## 5 0.77543680 -0.8795287 1.603651939 1.4894004 0.74063793 -3.0225778
## 6 0.37213224 -0.4033382 0.001081444 -0.0975633 -0.39245849 0.1715427
## lstat medv
## 1 0.8481491 -0.6401393
## 2 -0.8985906 0.6488122
## 3 -0.1153032 -0.1723628
## 4 -0.9751323 1.6344033
## 5 1.1687573 -1.0880779
## 6 -0.1643525 0.5733409
##
## Clustering vector:
## 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18
## 4 3 4 4 4 3 3 3 3 3 3 3 3 3 3 3 3 3
## 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36
## 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3
## 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54
## 3 3 3 2 2 2 3 3 3 3 3 3 3 3 3 3 2 2
## 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72
## 2 2 2 2 2 3 3 3 3 2 2 2 2 3 3 3 3 3
## 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90
## 3 3 3 3 3 3 3 3 2 3 2 3 3 3 3 3 4 4
## 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108
## 3 3 3 3 3 3 3 4 4 4 3 3 3 3 3 3 3 3
## 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126
## 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 1 3 3
## 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144
## 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 6 1
## 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162
## 1 1 1 1 1 1 1 1 6 1 6 6 5 4 3 1 6 4
## 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180
## 6 6 3 3 4 3 3 3 3 3 3 3 3 4 3 3 4 4
## 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198
## 4 4 4 4 3 3 4 2 2 2 2 2 2 2 2 2 2 2
## 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216
## 2 2 2 2 2 2 2 3 3 3 6 6 6 6 6 3 3 3
## 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234
## 6 3 6 6 6 6 6 4 4 4 4 4 4 4 3 4 4 4
## 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252
## 6 3 6 4 2 2 2 3 2 2 3 3 2 3 2 2 2 2
## 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270
## 2 2 2 2 2 4 4 4 4 4 4 4 4 3 4 4 4 6
## 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288
## 3 3 3 6 6 2 6 6 2 4 4 4 6 2 2 2 2 2
## 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 306
## 2 2 2 2 2 3 3 3 3 3 2 2 2 2 2 2 4 3
## 307 308 309 310 311 312 313 314 315 316 317 318 319 320 321 322 323 324
## 4 4 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3
## 325 326 327 328 329 330 331 332 333 334 335 336 337 338 339 340 341 342
## 3 3 3 3 3 3 3 2 2 3 3 3 3 3 3 3 3 2
## 343 344 345 346 347 348 349 350 351 352 353 354 355 356 357 358 359 360
## 3 2 2 3 3 2 2 2 2 2 2 2 2 2 6 6 6 1
## 361 362 363 364 365 366 367 368 369 370 371 372 373 374 375 376 377 378
## 1 1 1 6 6 1 1 5 1 6 6 1 6 1 1 1 1 1
## 379 380 381 382 383 384 385 386 387 388 389 390 391 392 393 394 395 396
## 1 1 5 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
## 397 398 399 400 401 402 403 404 405 406 407 408 409 410 411 412 413 414
## 1 1 1 1 1 1 1 1 5 5 1 1 1 5 5 5 5 5
## 415 416 417 418 419 420 421 422 423 424 425 426 427 428 429 430 431 432
## 5 5 5 5 5 5 1 1 1 5 5 5 5 5 5 5 5 5
## 433 434 435 436 437 438 439 440 441 442 443 444 445 446 447 448 449 450
## 5 5 5 5 5 5 5 1 1 1 1 1 1 5 1 1 1 1
## 451 452 453 454 455 456 457 458 459 460 461 462 463 464 465 466 467 468
## 5 1 1 1 5 5 5 5 1 1 1 1 1 1 1 1 5 1
## 469 470 471 472 473 474 475 476 477 478 479 480 481 482 483 484 485 486
## 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
## 487 488 489 490 491 492 493 494 495 496 497 498 499 500 501 502 503 504
## 1 1 1 1 1 1 1 3 3 3 3 3 3 3 3 3 3 3
## 505 506
## 3 3
##
## Within cluster sum of squares by cluster:
## [1] 725.2268 380.3152 712.0145 205.1638 359.7645 340.7321
## (between_SS / total_SS = 61.5 %)
##
## Available components:
##
## [1] "cluster" "centers" "totss" "withinss"
## [5] "tot.withinss" "betweenss" "size" "iter"
## [9] "ifault"Let’s use Elbow method to estimate number of clusters.
#Elbow Method for finding the optimal number of clusters
library(ggplot2)
set.seed(1234)
# Compute and plot wss for k = 2 to k = 15.
k.max <- 15
data <- boston_scaled1
wss <- sapply(1:k.max,
function(k){kmeans(data, k)$tot.withinss})
#wss
qplot(1:k.max, wss, geom = c("point", "line"), span = 0.2,
xlab="Number of clusters K",
ylab="Total within-clusters sum of squares")## Warning: Ignoring unknown parameters: span
## Warning: Ignoring unknown parameters: span
library(NbClust)
nb <- NbClust(boston_scaled1, diss=NULL, distance = "euclidean",
min.nc=2, max.nc=5, method = "kmeans",
index = "all", alphaBeale = 0.1)
## *** : The Hubert index is a graphical method of determining the number of clusters.
## In the plot of Hubert index, we seek a significant knee that corresponds to a
## significant increase of the value of the measure i.e the significant peak in Hubert
## index second differences plot.
## 
## *** : The D index is a graphical method of determining the number of clusters.
## In the plot of D index, we seek a significant knee (the significant peak in Dindex
## second differences plot) that corresponds to a significant increase of the value of
## the measure.
##
## *******************************************************************
## * Among all indices:
## * 12 proposed 2 as the best number of clusters
## * 6 proposed 3 as the best number of clusters
## * 3 proposed 4 as the best number of clusters
## * 3 proposed 5 as the best number of clusters
##
## ***** Conclusion *****
##
## * According to the majority rule, the best number of clusters is 2
##
##
## *******************************************************************#hist(nb$Best.nc[1,], breaks = max(na.omit(nb$Best.nc[1,])))Now, it’s much clearer that the data is described better with two clusters. With that, we run k-means algorithm again.
#Let us apply kmeans for k=4 clusters
km_final = kmeans(boston_scaled1, centers = 2) #we keep number of iter.max=15 to ensure the algorithm converges and nstart=50 to ensure that atleat 50 random sets are choosen
pairs(boston_scaled1[3:9], col=km_final$cluster)
More LDA
boston_scaled2<-as.data.frame(scale(Boston))
head(boston_scaled2)## crim zn indus chas nox rm
## 1 -0.4193669 0.2845483 -1.2866362 -0.2723291 -0.1440749 0.4132629
## 2 -0.4169267 -0.4872402 -0.5927944 -0.2723291 -0.7395304 0.1940824
## 3 -0.4169290 -0.4872402 -0.5927944 -0.2723291 -0.7395304 1.2814456
## 4 -0.4163384 -0.4872402 -1.3055857 -0.2723291 -0.8344581 1.0152978
## 5 -0.4120741 -0.4872402 -1.3055857 -0.2723291 -0.8344581 1.2273620
## 6 -0.4166314 -0.4872402 -1.3055857 -0.2723291 -0.8344581 0.2068916
## age dis rad tax ptratio black
## 1 -0.1198948 0.140075 -0.9818712 -0.6659492 -1.4575580 0.4406159
## 2 0.3668034 0.556609 -0.8670245 -0.9863534 -0.3027945 0.4406159
## 3 -0.2655490 0.556609 -0.8670245 -0.9863534 -0.3027945 0.3960351
## 4 -0.8090878 1.076671 -0.7521778 -1.1050216 0.1129203 0.4157514
## 5 -0.5106743 1.076671 -0.7521778 -1.1050216 0.1129203 0.4406159
## 6 -0.3508100 1.076671 -0.7521778 -1.1050216 0.1129203 0.4101651
## lstat medv
## 1 -1.0744990 0.1595278
## 2 -0.4919525 -0.1014239
## 3 -1.2075324 1.3229375
## 4 -1.3601708 1.1815886
## 5 -1.0254866 1.4860323
## 6 -1.0422909 0.6705582km_bs2<-kmeans(dist_eu, centers = 6)
#head(km_bs2)
myclust<-data.frame(km_bs2$cluster)
#myclust<-data.frame(km_bs2$cluster)
#myvar$boston_scaled2
myvar <- (myclust[,1])
#head(boston_scaled2)
lda.fit_bs2<-lda(myvar~., data = boston_scaled2 )
lda.fit_bs2## Call:
## lda(myvar ~ ., data = boston_scaled2)
##
## Prior probabilities of groups:
## 1 2 3 4 5 6
## 0.09486166 0.12845850 0.19169960 0.10079051 0.28260870 0.20158103
##
## Group means:
## crim zn indus chas nox rm
## 1 -0.3613809 -0.094199770 -0.47408693 1.5321752 -0.12487357 1.27068222
## 2 1.4172264 -0.487240187 1.06980230 0.4545202 1.34622349 -0.73713928
## 3 0.4194955 -0.487240187 1.14884305 -0.2723291 1.00261960 -0.29765790
## 4 -0.4149170 2.555355046 -1.22875891 -0.1951310 -1.21919439 0.78676843
## 5 -0.4055469 0.003897921 -0.72907965 -0.2723291 -0.78741061 0.07918728
## 6 -0.3559855 -0.464960891 0.08535629 -0.2723291 -0.03907908 -0.34955733
## age dis rad tax ptratio black
## 1 0.2307707 -0.3386056 -0.4961654 -0.7220694 -1.1226766 0.32813467
## 2 0.8557425 -0.9615698 1.2885597 1.2934457 0.4142248 -1.68787016
## 3 0.7547263 -0.7860200 1.2120560 1.2873665 0.6695878 0.03134501
## 4 -1.4488239 1.7464736 -0.7048880 -0.5692695 -0.8353442 0.34924852
## 5 -0.7882694 0.6799862 -0.5650500 -0.7231257 -0.2171966 0.37615865
## 6 0.4578808 -0.3069523 -0.5956710 -0.4102890 0.3497601 0.18939480
## lstat medv
## 1 -0.6138415 1.4407282
## 2 1.1961180 -0.8078336
## 3 0.6821488 -0.6015814
## 4 -0.9773530 0.8760790
## 5 -0.5789506 0.2036280
## 6 0.1782669 -0.3146198
##
## Coefficients of linear discriminants:
## LD1 LD2 LD3 LD4 LD5
## crim 0.06957743 0.29237657 -0.53832201 -0.50922053 -0.067791043
## zn -0.15484693 1.78015330 0.38305361 0.16031762 0.961795193
## indus 0.63817045 0.02644569 0.57679400 0.40269875 -0.080856440
## chas 0.15409184 0.15860914 -0.94342441 0.14850242 -0.003366811
## nox 1.27080209 0.78173714 -0.09726017 -0.26707799 -0.011605310
## rm -0.17566254 0.01368536 -0.06240662 0.73991062 -0.046102532
## age 0.11968517 -0.34671159 -0.07431380 0.27586717 0.910139703
## dis -0.32603841 0.55023745 0.12193551 -0.19596274 -0.582218617
## rad 0.78479581 -0.31767787 0.29044329 0.68534345 -1.614227061
## tax 0.65835846 1.01312767 0.24467220 0.08770833 0.563552653
## ptratio 0.31452906 0.11222333 0.30050298 -0.25310031 0.665512300
## black -0.31394272 -0.28807751 0.73424011 0.90913948 -0.145380180
## lstat 0.48061737 0.41720098 -0.52532223 0.20625897 0.101125672
## medv 0.24483789 0.59335240 -0.85238032 0.03044767 0.143223684
##
## Proportion of trace:
## LD1 LD2 LD3 LD4 LD5
## 0.7243 0.1511 0.0744 0.0292 0.0211# the function for lda biplot arrows
lda.arrows <- function(x, myscale = 1, arrow_heads = 0.1, color = "red", tex = 0.75, choices = c(1,2)){
heads <- coef(x)
arrows(x0 = 0, y0 = 0,
x1 = myscale * heads[,choices[1]],
y1 = myscale * heads[,choices[2]], col=color, length = arrow_heads)
text(myscale * heads[,choices], labels = row.names(heads),
cex = tex, col=color, pos=3)
}
plot(lda.fit_bs2, dimen = 2)
lda.arrows(lda.fit_bs2, myscale = 3)
Better ways to visualize LDA
library(plotly)##
## Attaching package: 'plotly'## The following object is masked from 'package:ggplot2':
##
## last_plot## The following object is masked from 'package:MASS':
##
## select## The following object is masked from 'package:stats':
##
## filter## The following object is masked from 'package:graphics':
##
## layoutmodel_predictors <- dplyr::select(train, -crime)
# check the dimensions
dim(model_predictors)## [1] 404 13dim(lda.fit$scaling)## [1] 13 3# matrix multiplication
matrix_product <- as.matrix(model_predictors) %*% lda.fit$scaling
matrix_product <- as.data.frame(matrix_product)
plot_ly(x = matrix_product$LD1, y = matrix_product$LD2, z = matrix_product$LD3, type= 'scatter3d', mode='markers', color = train$crim)Blog post by Jason Browniee on LDA
R-bloggers post on LDA R-bloggers post on K Means Clustering in R